Routh test

Question 1

A purely algebraic method for determining how many roots of the characteristic equation have positive real parts.

Answer 1

Routh test

Question 2

Case 1

Answer 2

If either one of the coefficients is negative, the system is unstable

Question 3

Case 2

Answer 3

If all coefficients are positive, the system maybe stable or unstable

Question 4

Used for Case 2

Answer 4

Routh array method

Question 5

Theorem 13.1

Answer 5

The necessary and sufficient condition for all the roots of the characteristic equation to have negative real parts (stable system) is that all elements of the first column of the Routh array be positive and nonzero.

Question 6

Theorem 13.2

Answer 6

If some of the elements in the first column are negative, the number of roots with a positive real part (in the right half-plane) is equal to the number of sign changes in the first column.

Question 7

Theorem 13.3

Answer 7

If one pair of roots is on the imaginary axis, equidistant from the origin, and all other roots are in the left half-plane, then all the elements of the nth row will vanish and none of the elements in the preceding row will vanish. The location of the pair of imaginary roots can be found by solving the equation